Heritability

If you take a trait like height, some people are tall, some people are tiny, i.e, height varies in a population. Genotype also varies in a population. People will have different alleles (or different combinations of alleles) to each other. Heritability attempts to measure how much of the variation in a trait is due to variation in genotype.

You don't really have additive and non-additive heritability but you do have additive and non-additive genetic effects.

Additive genetic effects can be though of as follows: given a phenotype and X number of alleles with an effect on that phenotype, additive simply means that the combined effect of each allele on a phenotype is equal to the sum of the individual effects.

Non-additive genetic effects are dominance and epistasis which do not function like this.

There are two kind of heritability, broad sense (H^2) and narrow sense (h^2).

Broad sense heritability (H^2) is difficult to estimate in natural populations like humans but can be done in experimental populations of model organisms. Broad-sense heritability measures the proportion of the variance (v) in a phenotype (P) that is attributable to all genotypic effects (vG) - where all genotypes includes additive, dominant and epistatic genotypes.

H^2 ~ vG/vP

Narrow sense heritability (h^2) can be estimated from twin studies and is concerned only with the proportion of phenotypic variance that is due to additive genotypic effects (Ga).

h^2 ~ vGa/vP

So, when you hear studies talking about the heritability of disease risk or height or leadership or anything, this is what they are referring to.

Hope that helps.

Note, when you say phenotype influenced, what do you mean?